Difference between revisions of "Polar correspondence"
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| + | A correspondence between two surfaces such that at corresponding points the radius vector of one of them is parallel to the normal of the other, and vice versa. For every smooth surface $ F $ | ||
| + | in $ E ^ {3} $ | ||
| + | with radius vector $ x $ | ||
| + | there exists (under certain conditions) a surface $ F ^ { * } $ | ||
| + | polar with it and with radius vector $ x ^ {*} = - n / ( x , n ) $, | ||
| + | where $ n $ | ||
| + | is the normal and $ ( x , n ) $ | ||
| + | is the support function to $ F $, | ||
| + | so that | ||
| + | |||
| + | $$ | ||
| + | ( x ^ {*} , x ) = 1 ,\ ( x _ {i} , x ) = \ | ||
| + | ( x _ {i} , x ^ {*} ) = 0 . | ||
| + | $$ | ||
Sometimes these conditions are also included in the definition of a polar correspondence. | Sometimes these conditions are also included in the definition of a polar correspondence. | ||
The concept of polar correspondence shows itself particularly clearly (in the sense of a complete duality) in centro-affine geometry. | The concept of polar correspondence shows itself particularly clearly (in the sense of a complete duality) in centro-affine geometry. | ||
Latest revision as of 08:06, 6 June 2020
A correspondence between two surfaces such that at corresponding points the radius vector of one of them is parallel to the normal of the other, and vice versa. For every smooth surface $ F $
in $ E ^ {3} $
with radius vector $ x $
there exists (under certain conditions) a surface $ F ^ { * } $
polar with it and with radius vector $ x ^ {*} = - n / ( x , n ) $,
where $ n $
is the normal and $ ( x , n ) $
is the support function to $ F $,
so that
$$ ( x ^ {*} , x ) = 1 ,\ ( x _ {i} , x ) = \ ( x _ {i} , x ^ {*} ) = 0 . $$
Sometimes these conditions are also included in the definition of a polar correspondence.
The concept of polar correspondence shows itself particularly clearly (in the sense of a complete duality) in centro-affine geometry.
How to Cite This Entry:
Polar correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar_correspondence&oldid=48227
Polar correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar_correspondence&oldid=48227
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article